# Similar Shapes, calculating the length.

The questions I'll be asking today are in regards to my math exam which will be coming up next Friday. I have been working through past exam papers however have run into a couple of issues, and being the weekend, and a teacher only day on Monday, I am unable to ask questions for the next three days. I really appreciate all the help you will be able to give me, and will show some working/my understanding of questions, in order to show that I have attempted the question, not just wanting you to do my homework

The two questions below are in regards to similar shapes. I will write the question, then my understanding.

1) Triangles ACE and BCD are similar. BD is parallel to AE.

AE is 10cm , BD is 6CM , AC is 7CM.

Calculate the length of BC

I'm having massive issues with this question, and don't know where to start. Are you able to give me a worked formula?

2) The sketch below shows an ice cream cone of a radius of 4cm and depth of 12cm in which there is a sphere of ice cream of radius 2.5cm. The ice cream melts and runs into the base of the cone. Find the depth of the liquid ice cream when this has happened. YOU MUST USE THE SIMILAR SHAPES METHOD

This is what my understanding is so far.

We first calculate the volume of the ice cream sphere. So $\frac {4} {3} \times π \times r^2. Therefore,\frac {4} {3} \times π \times 2.5 \times 2.5 = 26.18cm^3$.

Then we calculate the volume of the cone, = $201.06cm^3$.

How would we go about finding the depth?

**I appreciate EVERYONE's help in answering this. You probably know how nerve racking exams are, so a worked formula would be so much appreciated. I really really really do appreciate it! **

• Let volume of cone up to a certain height "h" with which it is filled with volume occupied of sphere. – Mann May 9 '15 at 6:06

Remember,similar shape is about comparing the scale factor or ratio.

This is for Q1 solution though I just learned similar triangle in grade 8 Singapore.This is the ratio you SHOULD know for similar triangle AC:BC=AE:BD

We have length of AE and BD so we simplify the ratio which is 5:3

Since the ratio of AC:BC=AE:BD,and we know that AC=10cm,we can now solve BC which is

BC=>$\frac {10} {5} \times 3 =6cm$

This is my ATTEMPT for Q2.

Same thing as before,we just compare the ratio of the radius of the 2.The ratio of sphere radius and the larger cone radius is 2.5:4 which is simplified to 5:8 (Best to have an integer ratio)

Q2:Ratio of longer radius and shorter radius=>5:8 Now we can form an equation by linking 2 volume of cones together to this.(I actually never learned the volume of sphere and cones :/But for my guess,the formula for cone is $πr^2 \frac {h} {3}$)

Thanks to @AndréNicolas, this is the ratio.

$\frac {π} {3} (4^2) (12)$(Volume of cone):$\frac {4} {3} π (2.5)^3$(Volume of sphere)

From here, we get $\frac {125} {384}$ as our scale factor.

Finally, (12)($\sqrt [3] {\frac {125} {384}}$)=8.254818122...cm

OR,

Using similar triangles as stated by @mann,we have a ratio of 8:5

so,$\frac {12} {8} \times 5$=7.5cm

So I have 2 possible answer.I will see which one is right later

• Hint: $h=r \left(\frac{2H}{R}\right)^{2/3}$ , r= radius of sphere , H= Height of ice cream cone. , R= Radius of ice cream cone. – Mann May 9 '15 at 6:58
• Wow, I really appreciate your thorugh response! Which one would be correct however? – Alex May 9 '15 at 7:47
• @Alex I am learning similar shapes too and if I were myself,my answer would be the one with 7.5cm (Apologies,I still couldn't decide which answer is correct but I do not think you are learning the complicated one,so I suggest the answer with 7.5cm,unless you have learned linear,area,volume scale.)For the first question I am sure my answer is correct – ministic2001 May 9 '15 at 7:52
• I really really appreciate everyone's assistance on this!! – Alex May 9 '15 at 8:02

Hint for Q2:

As you can see $r\neq r_0$
But volume of the sphere and volume cone filled are same , later on you can use similarity of triangles to make two equation and arrive at your result.

• I have done my attempt for question 2.Is my question 2 correct? – ministic2001 May 9 '15 at 6:22
• You should use $\frac{4}{3}\pi r^3 = \frac{1}{3} \pi r_0^2 h$ and $\frac{r_0}{h}=\frac{R}{H}$ @ministic2001 – Mann May 9 '15 at 6:25
• Oh so thats the formula for sphere.Re-edit for mine!I am learning at the same time :D – ministic2001 May 9 '15 at 6:41
• I am glad :D , you see the volume here is conserved, as stated in question the sphere melts goes into the ice cream cone and molds itself into a cone who's radius or height we don't know. However we knew it's volume and it must be same as sphere unless of course things have evaporated or leaked :) – Mann May 9 '15 at 6:43
• Actually,I kinda understood what he actually meant.It is advanced,I can see..All I need to do seems to be converting the ratio of the volume and put ad "P" and then substitute "P" into your (12)($\sqrt [3] {p}$)I'm just a grade 8 student though :/ – ministic2001 May 9 '15 at 7:03